Does the decreasing sequence $Ma\supseteq Ma^2\supseteq\ldots\supseteq Ma^n\supseteq\ldots$ always stabilise when $R$ is artinian?

Question

Let $R$ be a commutative artinian ring with unity, $M$ be an $R$-module and $a\in R$. Is it always true that the decreasing sequence $$Ma\supseteq Ma^2\supseteq\ldots\supseteq Ma^n\supseteq\ldots$$ must stabilise for some $n\in \Bbb Z_{>0}$?

My thinking is that it is TRUE, but the reasoning is not that standard!

Since $R$ is artinian, $R$ has d.c.c. on the principal ideal $aR$ generated by $a\in R$. That is, $aR\supseteq a^2R\supseteq\ldots\supseteq a^nR= a^{n+1}R=\ldots$ stabilises for some $n>0$. This passes over to $Ma\supseteq Ma^2\supseteq\ldots\supseteq Ma^n= Ma^{n+1}=\ldots$ stabilises for some $n>0$.

Answer 1

Yes. Perhaps it looks better if you first show $Ma^i=M(a^iR)$, because that makes it a little more obvious why the chain of ideals is connected.